Actually...this is not entirely true. Not only does the president not have such powers but much of the power that the president
does have in this realm is without constitutional basis (these powers are inferred as a result of precedents set by several presidents throughout history). According to historian Thomas Woods, "Ever since the Korean War, Article II, Section 2 of the Constitution — which refers to the president as the 'Commander in Chief of the Army and Navy of the United States' — has been interpreted to mean that the president may act with an essentially free hand in foreign affairs, or at the very least that he may send men into battle without consulting Congress."
Here's what the constitution says, re: the president as commander in chief:
The President shall be Commander in Chief of the Army and Navy of the United States, and of the Militia of the several States, when called into the actual Service of the United States; he may require the Opinion, in writing, of the principal Officer in each of the executive Departments, upon any Subject relating to the Duties of their respective Offices, and he shall have Power to grant Reprieves and Pardons for Offenses against the United States, except in Cases of Impeachment.
Bringing the troops back home is not only not solely a presidential responsibility (either officially or precedentially) but it clashes entirely with American Interests.